I don't buy the "wig sticking to the inside of the helmet" story. Couldn't they use stronger adhesive for the hair on the head prosthetic? Or how about doing an unmasking scene with a specifically made helmet that's missing its back half so the hair doesn't get stuck to it. You can hide the missing gap in the helmet so it looks like an ordinary helmet removal.
But worst of all, what about the lack of facial hair? That's what stinks about this excuse from the makeup department. It may be a stereotype, but Klingons always rock the facial hair. I'm sure I've even seen some female Klingons with facial hair in at least one episode. The fact that the unmasked Klingon is clean shaven leads me to conclude that this is not about a makeup malfunction. I imagine this little spiel was presented by someone in Abrams' office at some point during preproduction:
"We're taking the Trek mythos in a new and exciting direction! First bald Romulans, now bald Klingons! It's a pity we can't make Cumberbatch bald for the film too, but we can't afford to sweep another lawsuit under the rug after he defenestrated the last stylist who tried to shave him against his will. Love that guy! He'll be a perfect Khan!"